Theta function of an ideal

Question

While attempting to find the functional equation for the dedekind zeta function, I encountered this function: $\theta(x_1,x_2,...,x_{r_1+r_2})=\sum_{\alpha \in I_c} e^{-\pi \sum_{k=1}^{r_1+r_2}x_k \mid \sigma_k(\alpha) \mid^2}$ Where $I_c$ is an ideal in a class group, $\sigma_k$ are the embeddings of the field, and $r_1+r_2$ is the rank of the unit group+1. Does this function obey a functional equation? Or is there a similar function that does? Thanks for any help.

Answer 1

$I_c$ is a lattice. Fix $a_j$ then for some matrix $M = M(a,c)$ $$\theta(a_1 t,\ldots a_n t) = \sum_{l \in \Bbb{Z}^n} e^{- \pi t \| Ml \|^2} = \sum_{k \in \Bbb{Z}^n} t^{-1} e^{- \pi t^{-1}\| M^{-\top} k \|^2}$$ (Poisson summation formula, Fourier series, Fourier transform of the Gaussian..)

The functional equation of $\zeta_K(s)$ is obtained from that $M^{-\top}$ corresponds to what we'd get when replacing $(a_1,\ldots,a_n), I_c$ by $(a_1^{-1},\ldots,a_n^{-1}), I_{c^{-1}}$